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## Fermions with repulsive interactions + Multi-Orbital Quantum Phase Diffusion

Chan, Cecilia, Peoria City Hall Reporter has reference to this Academic Journal, PHwiki organized this Journal Interacting Bosons in addition to Fermions in 3D Optical Lattice Potentials Sebastian Will, Thorsten Best, Simon Braun, Ulrich Schneider, KC Fong, Lucia Hackermüller, Stefan Trotzky, Yuao Chen, Ute Schnorrberger, Stefan Kuhr, Jacob Sherson, Christof Weitenberg, Manuel Endres, Theory: Belén Paredes, Mariona Moreno Immanuel Bloch Johannes Gutenberg-Universität, Mainz funding by DFG, European Union, $ AFOSR, DARPA (OLE) www.quantum.physik.uni-mainz.de Fermions in a 3D Lattice with repulsive interactions Quantum Phase Diffusion in addition to Bose-Fermi Mixtures Our starting point: Ultracold Quantum Gases Bose-Einstein condensate e.g. 87Rb atoms ground states at T=0 Parameters: Densities: 1015 cm-3 Temperatures: nanoKelvin Number of Atoms: about 106 Degenerate Fermi gas e.g. 40K atoms

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Optical Lattice in 3D 3D lattice: array of quantum dots Hubbard Hamiltonian Restriction to single (lowest) b in addition to in addition to expansion in localized wannier functions yields: Tunneling matrix element: Onsite interaction matrix element: Bose- Hubbard Hamiltonian Fermions with repulsive interactions U. Schneider, L. Hackermüller, S. Will, Th. Best, I.Bloch & A. Rosch, Th. Costi, D. Rasch, R. Helmes (Science, 322, 1520 (2008))

Strongly Interacting Fermions in Optical Lattices Phases predicted at half filling as long as strong interactions U/12J > 1: Related experimental work at ETHZ (T. Esslinger) e.g. M. Köhl et al., PRL 94, 080403 (2005), R. Jördens et al. Nature 455, 204 (2008) maximal entropy: S/N = kB 2 ln(2) Hubbard Model in addition to High-Tc Can we help identifying the phase diagram of the Hubbard model W. Hofstetter, J.I. Cirac, P. Zoller, E. Demler, M.D. Lukin, PRL 89, 220407 (2002), P. A. Lee, N. Nagosa, X. G. Wen, Rev. Mod. Phys. 78 , 17 (2008) Experimental Setup: Fermions in the Optical Lattice Spin mixture of K atoms in F=9/2, mF=-9/2 in addition to F=9/2, mF=-7/2: Crossed Dipole Trap 1030nm (elliptical beams) Blue Detuned Lattice Beams 738nm (160 µm waist) T=0.06 to 0.13 TF with about 3 x 105 atoms!

Compression of the Quantum Gas Total Potential as long as Atoms: Optical Lattice combined with Dipole Trap! Compression Range: + Hubbard Hamiltonian: All Parameters Tunable! 40K Feshbach resonance: + (JILA parametrization) Experimental Observables: Global Observable: Compressibility Local Observable: For example: in-situ cloud size with phase-constrast imaging For example: pair fraction with Feshbach ramp or central occupation (see L. De Leo et al., 2008, alternative method: see Zürich experiment R. Jördens et al., Nature 455, 204 (2008)) U. Schneider, L. Hackermüller, S. Will, Th. Best, I.Bloch & A. Rosch, Th. Costi, D. Rasch, R. Helmes (Science, 322, 1520 (2008)) 2R

Quantum Phases of Repulsive Fermions in Trap compressible! incompressible! incompressible! Comparison with Theory (I) Dynamical Mean Field Theory (DMFT) Metzner, Vollhardt, Georges, Kotliar e.g. A. Georges et al. Rev. Mod. Phys. 68, 13 (1996) Real Space Adaptation (Inhomogeneous Systems) Achim Rosch, Theo Costi (here LDA + DMFT) see also: L. De Leo et al. PRL, 101, 210403 (2008) in addition to work by W. Hofstetter Calculations at Forschungszentrum Jülich: JUGENE, IBM Blue-Gene Supercomputer 1 in Germany 6 on TOP 500 list worldwide First test bed as long as DMFT in 3D! Measuring the Cloud Size

Measuring the Compressibility (I) U. Schneider, et al. (Science, 322, 1520 (2008)) Theory: R. W. Helmes et al. (PRL 100, 056403(2008)) Measuring the Compressibility (II) Pair Fraction versus Compression

Entropy Distibution in the Trap Entropy of non-interacting gas in harmonic trap T/TF = 0.15 S/N > kB 2 ln(2) While entropy of MI is only S/N = kB ln(2) ! U/12J = 1.5 Summary: Pair Fraction & Compression Measurements In-situ cloud size / Compression Measurements: Pair fraction measurements: Very good quantitative agreement with ab-initio DMFT as long as weak in addition to strong compressions! Direct measurement of the (in-)compressibility of the many-body system. Deviations beyond U/6J = 4 in low compression regime Good agreement with ab-initio DMFT theory (T approx. 0.15 TF) But note: Melted MI in addition to strongly interacting metallic phases can also show suppression of pairs! Multi-Orbital Quantum Phase Diffusion Sebastian Will, Thorsten Best, Simon Braun, Ulrich Schneider, KC Fong, Lucia Hackermüller, Dirk-Sören Lühmann, Immanuel Bloch

From BEC to a Superfluid in an Optical Lattice BEC in a harmonic trap Onsite picture: Coherent State Poisson distribution Non-interacting, homogeneous case: plus a weak lattice Dynamics of a coherent state: In the limit of zero tunneling (J = 0) evolution is determined by: The matterwave field on a lattice site experimentally observable as time-evolution of coherent state Visibility Phase Diffusion Dynamics: Collapse in addition to Revival Matterwave field collapses in addition to revives after multiple times of h/U Collapse time depends on the variance of the atom number distribution Theory: Yurke & Stoler, 1986, F. Sols 1994; Wright et al. 1997; Imamoglu, Lewenstein & You et al. 1997, Castin & Dalibard 1997, E. Altman & A. Auerbach 2002, Exp: M. Greiner et al. 2002, G.-B. Jo et al. 2006, J. Sebby-Strabley et al. 2007, A. Widera et al., 2007, M. Oberthaler et al. 2008

Dynamical Evolution of the Interference Pattern t=50µs t=150µs t=200µs t=300µs t=400µs t=450µs t=600µs Dynamics after potential jump from 8Erec to 22Erec! Collapse & Revival under Optimal Harmonic Confinement Up to 70 revivals can be detected! And: Multiple frequency components! Why Multiple Frequencies Here U is assumed to be constant, independent of filling Breakdown of the single-b in addition to approximation! Admixture of higher-b in addition to orbitals! as long as a differential measurement, see also: G. Campbell et. al., Science (2006)

Fourier Spectrum Strong signal of small contributions due to heterodyning effect! E(2) + E(4) 2E(3) 2 E(2) – E(3) E(2) c2 · c32 · c4 c1 · c22 · c3 c0 · c12 · c2 Comparison with Exact Diagonalization Theory: D.-S. Lühmann, Hamburg University of order 2U of order U Atom distribution along the SF to MI transition:

Conclusion Global Compressibility Measurements on Repulsively Interacting Fermi Gases in a 3D Optical Lattice Evidence as long as Incompressible Mott Core Good Agreement with ab-initio DMFT calculations Quantum Phase Diffusion as a Probe in Strongly Interacting Quantum Gas Mixtures Quantum Phase Diffusion with Fock State Resolution Renormalized Hubbard Parameters Self-Trapping in Bose-Fermi Mixtures (Multi-B in addition to Physics) THANK YOU! Useful Variables: Interactions versus Kinetic Energy Confinement versus Kinetic Energy, where Initial Temperature (Entropy) characteristic trap energy = Fermi energy at T=0, J=0 in addition to no interaction trap aspect ratio: Doubly occupied sites, compressibility, ,

## Chan, Cecilia Peoria City Hall Reporter

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